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Izv. RAN. Ser. Mat., 1996, Volume 60, Issue 2, Pages 159–194 (Mi izv75)  

This article is cited in 6 scientific papers (total in 6 papers)

Cycles on Abelian varieties and exceptional numbers

S. G. Tankeev

Vladimir Technical University

Abstract: The article considers a technique for proving the Hodge, Tate, and Mumford–Tate conjectures for a simple complex Abelian variety $J$ of non-exceptional dimension under the condition that $\operatorname{End}(J)\otimes \mathbb R\in\{\mathbb R,M_2(\mathbb R), \mathbb K,\mathbb C\}$, where $\mathbb K$ is the skew field of classical quaternions. The simple $2p$-dimensional Abelian varieties over a number field ($p$ is a prime, $p\geqslant 17$) are studied in detail. An application is given of Minkowski's theorem on unramified extensions of the field $\mathbb Q$ to the arithmetic and geometry of certain Abelian varieties over the field of rational numbers.

DOI: https://doi.org/10.4213/im75

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English version:
Izvestiya: Mathematics, 1996, 60:2, 391–424

Bibliographic databases:

UDC: 512.6
MSC: Primary 14K15, 14C30; Secondary 17B10
Received: 25.04.1995

Citation: S. G. Tankeev, “Cycles on Abelian varieties and exceptional numbers”, Izv. RAN. Ser. Mat., 60:2 (1996), 159–194; Izv. Math., 60:2 (1996), 391–424

Citation in format AMSBIB
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\by S.~G.~Tankeev
\paper Cycles on Abelian varieties and exceptional numbers
\jour Izv. RAN. Ser. Mat.
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\vol 60
\issue 2
\pages 159--194
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. S. G. Tankeev, “On Frobenius traces”, Izv. Math., 62:1 (1998), 157–190  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    2. S. G. Tankeev, “Cycles of small codimension on a simple $2p$- or $4p$-dimensional Abelian variety”, Izv. Math., 63:6 (1999), 1221–1262  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    3. S. G. Tankeev, “On the standard conjecture for complex Abelian schemes over smooth projective curves”, Izv. Math., 67:3 (2003), 597–635  mathnet  crossref  crossref  mathscinet  zmath  isi
    4. S. G. Tankeev, “On the numerical equivalence of algebraic cycles on potentially simple Abelian schemes of prime relative dimension”, Izv. Math., 69:1 (2005), 143–162  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    5. Vasiu A., “Some cases of the Mumford-Tate conjecture and Shimura varieties”, Indiana University Mathematics Journal, 57:1 (2008), 1–75  crossref  mathscinet  zmath  isi  scopus
    6. Yu Ch.-F., “a Note on the Mumford-Tate Conjecture For Cm Abelian Varieties”, Taiwan. J. Math., 19:4 (2015), 1073–1084  crossref  mathscinet  zmath  isi  elib  scopus
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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